Optical fiber based on wireless scheme for wideband multimedia access

ABSTRACT

A Fiber-wireless uplink consists of a wireless channel followed by a radio-over-fiber (ROF) link. Typically, nonlinear distortion of the ROF link is the major concern when the radio frequency is only a few GHz. This especially severe in the uplink, because of the multipath fading of the wireless channel. A Hammerstein type decision feedback equalizer is described for the fiber wireless uplink, that compensates for nonlinear distortion of the ROF link as well as linear dispersion of the wireless channel. Since the linear and nonlinear parts of the receiver are separated, tracking the fast changing wireless channel is virtually independent of compensating for the relatively static nonlinearity. Analytical results show that the receiver provides excellent compensation with notably less complexity.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority from U.S. provisional No.60/301,170 filed Jun. 28, 2001.

BACKGROUND OF THE INVENTION

[0002] Optical fiber based wireless access schemes have become verypopular recently because of their potential to increase system capacity,enable wideband access and to cover special areas such as tunnels andsupermarkets. These schemes are especially useful for indoorapplications with micro and pico cellular architecture.

[0003] When the fiber is short (say, less than a few kilometers) and theradio frequency is only a few GHz, effects of fiber dispersion and laserchirp are negligible [1]. All cited references are listed at the end ofthis patent disclosure. This is especially true at 1310 nm. Therefore,the ROF link has adequate bandwidth to support wireless multimediaservices. In this case, however, nonlinear distortion of the electricalto optical conversion process becomes the major limitation. Theimpairment is severe in the uplink where, the received signal largelyfluctuates due to multipath fading of the wireless channel. Both directmodulation and external modulation schemes suffer from limited dynamicrange because of this nonlinear distortion. We focus on AM-AM and AM-PMtype nonlinear distortion considering the whole ROF link. However,clipping is neglected.

[0004] There have been several attempts to increase the linearity of theROF link by fixed electronic means [2]. However, fixed schemes sufferfrom device dependency. We did some work focusing on adaptivecompensation of the ROF link nonlinearity, assuming an AWGN wirelesschannel [3]. However, for a realistic solution wireless channel fadinghas to be considered also. Recently, we reported an algorithm toidentify both the dispersive wireless channel and the nonlinear fiberchannel [4].

[0005] The nonlinearity of the ROF link contributes to various kinds ofimpairment. When the negative peak of the modulating signal goes belowthreshold level, a clipping distortion occurs. In a multicarrierenvironment, intermodulation distortion also occurs. In addition, evenwith a single carrier and no clipping, the mild in-band nonlinearitycauses AM-AM and AM-PM type nonlinear distortion.

[0006] There have been several attempts to increase the linearity of theROF link by fixed electronic means. These are discussed in detail in[5]. Recently, some work has been done focusing on adaptive compensationof the ROF link nonlinearity, assuming an AWGN wireless channel [3],[6]. However, a good estimation of not only the nonlinearity, but alsothe multipath wireless channel is essential for efficient equalization.

[0007] The fiber-wireless uplink can be modeled as a Wiener system.Reputed mathematician N. Wiener first showed that any BIBO stablenonlinear system with finite memory can be modeled as a Wiener systemfor Gaussian inputs which, consists of orthogonal linear dynamicfunctions followed by static nonlinear functions [7].

[0008] However, due to the practical difficulties in generating Gaussianinputs, different approaches have been proposed. Pseudorandom (PN)sequences have white noise like properties and, easy to generate andanalyze. Their correlation properties are well understood [8]. Besides,maximal length PN sequences are widely used in spread spectrumcommunications. Therefore, using PN sequence for channel estimation isvery attractive in wireless communications. Billings and Fakhouriinitially used PN sequences for control system identification [9].

SUMMARY OF THE INVENTION

[0009] This invention provides a Hammerstein type decision feedbackequalizer (HDFE) for use in a Wiener type communications system, as forexample the fiber wireless uplink of a radio over fiber (ROF) wirelesscommunications system. The HDFE compensates separately for nonlineardistortion of the ROF link as well as linear dispersion of the wirelesschannel. Since the linear and nonlinear parts of the receiver areseparated, tracking the fast changing wireless channel is virtuallyindependent of compensating for the relatively static nonlinearity.Analytical results show that the receiver provides excellentcompensation with notably less complexity.

[0010] The invention finds use in a central base station of acommunications network, wherein the central base station communicatesover a channel with multiple portable units, the channel having achannel impulse response and the channel contributing linear andnonlinear distortion to a signal x(n) transmitted over the channel. AnHDFE comprises a polynomial filter for receiving signals from theportable units and for producing an output signal z(n) that iscompensated for nonlinear channel distortion; and a filter sectionfollowing the polynomial filter for compensating for linear channeldistortion. The polynomial filter is preferably memoryless andconfigured to apply a polynomial having polynomial coefficients g_(i)calculated from the signals received from the portable units, preferablywithout explicitly estimating a polynomial that characterizes thechannel.

[0011] Preferably, the equalizer is configured to calculate thepolynomial coefficients gi by:

[0012] estimating the channel impulse response h(n), preferably by usingcorrelation properties of PN sequences, and, even more preferably usinga Vandermonde matrix approach with projection of linear and nonlinearcoefficients of the feedforward filter and the feedback filter ontodifferent subspaces;

[0013] correlating the channel impulse response h(n) with the signalx(n) transmitted over the channel to produce a signal q(n); and

[0014] selecting the coefficients g_(i) so that the output z(n) of thepolynomial filter is proportional to q(n).

[0015] The coefficients of the polynomial may be selected by a methodfrom the group consisting of QR decomposition and an adaptive technique.The filter section is preferably a decision feedback equalizer having afeedforward filter and a feedback filter.

[0016] These and other aspects of the invention are described in thedetailed description of the invention and claimed in the claims thatfollow.

BRIEF DESCRIPTION OF THE DRAWINGS

[0017] There will now be described preferred embodiments of theinvention, with reference to the drawings, by way of illustration onlyand not with the intention of limiting the scope of the invention, inwhich like numerals denote like elements and in which:

[0018]FIG. 1 shows an optical fiber based wireless access scheme tosupport wireless multimedia access and to increase the capacity;

[0019]FIG. 2 shows a schematic of a fiber-wireless uplink with theHammerstein type DFE;

[0020]FIG. 3 is a graph showing higher order expectations of the channelimpulse response;

[0021]FIG. 4 is a graph showing mean squared error of the polynomialfilter Vs SNR as a function of channel memory L;

[0022]FIG. 5 is a graph showing BER performance of the HDFE and astandard DFE;

[0023]FIG. 6 is a schematic showing a Fibre-Wireless Uplink;

[0024]FIG. 7 is a block diagram showing a simulation model of the methodof channel estimation according to an embodiment of the invention;

[0025]FIG. 8 is a graph showing actual and estimated impulse responsesfor a four path impulse response;

[0026]FIG. 9 is a graph showing actual and estimated impulse responsesfor an eight path impulse response; and

[0027]FIG. 10 is a graph showing received data and an estimatedpolynomial using an embodiment of the invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

[0028] In this patent document, “comprising” means “including”. Inaddition, a reference to an element by the indefinite article “a” doesnot exclude the possibility that more than one of the element ispresent.

[0029] Referring to FIG. 1, there is shown an optical fiber basedwireless scheme for wideband multimedia access. A central base station10 communicates over radio over fiber links 12 with a number of radioaccess points 14. Each radio access point 14 serves portable wirelessunits 16 in a micro-cell or pico-cell 18 whose footprint may be in theorder of tens of meters.

[0030] According to the invention, a Hammerstein type decision feedbackequalizer (HDFE) is used for the fiber-wireless uplink. This receiverindividually compensates for both nonlinear distortion and the timedispersion. It has a unique architecture with notably less complexity,nevertheless its performance is close to the performance of a standardDFE in a linear channel.

[0031] The decision Feedback equalizer (DFE) has been very successful inwireless communications, thanks to its robustness in mitigating commonlyencountered spectral nulls in frequency selective fading channels.Strictly speaking the basic DFE itself is nonlinear because of thedecision device and the feedback loop [10]. However, it is important tonote that, although the basic DFE is nonlinear, it is effective inequalizing only linear channels. Besides, both the feedback and the feedforward filters in the basic DFE are linear. When the channel itself isnonlinear with AM-AM and AM-PM type distortions, the basic DFE has to beenhanced in some way to accommodate the additional nonlinearity.

[0032] Mathematically, an equalizer should have an exact inverse of thechannel structure. Referring to FIG. 2, the fiber-wireless uplink 20consists of a linear dynamic system 22 (the wireless channel) followedby a static nonlinear system 24 (the optical channel). Therefore, it canbe modeled as a Wiener system. The inverse of a Wiener system is aHammerstein system. Furthermore, the fiber-wireless channel has thefollowing properties:

[0033] 1. The wireless channel 22 varies relatively fast. This impliesthat the compensation should follow it in real time.

[0034] 2. The nonlinearity comes from a laser diode (not shown) and froman RF amplifier (not shown) in the fiber-wireless uplink 20. Hence, itis almost stationary. That means the nonlinear compensation needs to beupdated only occasionally.

[0035] From the foregoing, an efficient equalizer should separatelycompensate for linear and nonlinear distortions. A receiver 26 thataccomplishes this intention is shown in FIG. 2 including a polynomialfilter (PLF) 28 for receiving signals from the portable units 16 and forproducing an output signal z(n) that is compensated for nonlinearchannel distortion and a filter section 30 following the polynomialfilter 28 for compensating for linear channel distortion. PLF 28 is amemoryless polynomial filter of order N. The filter section 30 includesa linear feed forward filter 32 (FFF) and a linear feedback filter 34(FBF), both with memory. This is a novel receiver architecture for thefiber-wireless uplink 20 where, an inverse of a polynomial implementedby the PLF 28 models the optical link nonlinearity while the restcompensates for the wireless channel dispersion.

[0036] As shown in FIG. 2, x(n) is the transmitted data from theportable unit; h(n) is the impulse response of the (linear) wirelesschannel 22; q(n) is the signal received at remote antennas 15 at theradio access points 14. The antennas 15 are connected to conventionaloptical modulators (not shown) on the fibers 12. Note that q(n) is aninternal signal that is not accessible. At the central base station 10,the optical signal is converted back to electrical signal which, wedenote as r(n). This r(n) is available at the receiver. The nonlineartransfer function of the complete optical link, denoted by F(.), mapsq(n) to r(n) so that, r(n)=F[q(n)]+v(n). All the electrical and opticalnoise is appropriately transferred to the input of the receiver anddenoted by v(n). The optimization of the DFE involves the selection ofthe parameters of the polynomial filter 28 as well as of the linearfilters 32, 34 in the filter section 30.

[0037] Referring to FIG. 2, the output of the polynomial filter z(n) hasthe form,

z(n)=g ₀ +g ₁ r(n)+g ₂ r ²(n)+ . . . g _(N) r ^(N)(n)  (1)

[0038] There are no delay terms since we assume no memory for thefilter. The weights g_(i) (0≦i≦N) have to be determined such that, thispolynomial is an inverse of the channel nonlinearity F(.). Equivalently,z(n) should be made proportional to q(n).

[0039] The inverse polynomial is preferably generated from the receivedsignal r(n) without explicitly estimating the channel polynomial. Thisis accomplished as follows: first, the linear channel impulse responseh(n) is estimated as described below. Then, q(n) is determined byconvolving the given x(n) with this h(n). Knowing both q(n) and r(n),the output z(n) can be made proportional to the q(n) by appropriatelyselecting each g_(i). This can be done for example either by a standardQR decomposition method or by adaptive techniques as shown for examplein [14].

[0040] In a QR decomposition method, first, r(n) is expanded into aVandermonde matrix Vr of N_(L) rows and (N+1) columns. Here, N_(L) isthe number of sample points in r(n). $\begin{matrix}{V_{r} = \begin{bmatrix}1 & {r(1)} & {r^{2}(1)} & \ldots & {r^{N - 1}(1)} & {r^{N}(1)} \\1 & {r(2)} & {r^{2}(2)} & \ldots & {r^{N - 1}(2)} & {r^{N}(2)} \\1 & \ldots & \ldots & \ldots & \ldots & \ldots \\\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\1 & {r\left( N_{L} \right)} & {r^{2}\left( N_{L} \right)} & \ldots & {r^{N - 1}\left( N_{L} \right)} & {r^{N}\left( N_{L} \right)}\end{bmatrix}} & (2)\end{matrix}$

[0041] A vector q and a vector G are defined as,

[0042] q[q(1)q(2)q(3) . . . q(N_(L))]^(T)

[0043] G=[g₀g₁g₂ . . . g_(N)]^(T)

[0044] Now, the target is to determine the weights of the vector G suchthat, V_(r)G=z where, z is an estimate of q. An efficient way ofcomputing G is to perform an orthogonal-triangular decomposition of theVandermonde matrix V_(r) such that it can be ten as:

V _(r) =Q _(r) R _(r)  (3)

[0045] Now, from the properties of R_(r) and Q_(r), we have

R _(r) G=Q _(r) ^(T) Z  (4)

[0046] Note that Q_(r) ^(T)z is a vector of length (N+1). Since R_(r) isa triangular matrix, this equation can be efficiently solved for G byback substitution. The optimal filter order N is selected by increasingN iteratively until the error ε_(p)=q−z is sufficiently small.

[0047] Weights of both the F F F and F B F are optimized jointly.Referring to FIG. 2, define the combined data input vector to the linearfilters as, $\begin{matrix}{{{U_{L} = {\left\lbrack {z^{T}x^{T}} \right\rbrack^{T}\quad {where}}},{z = {\left\lbrack {{z\left( {{- N_{f}} + 1} \right)}{z\left( {{- N_{f}} + 2} \right)}\quad \ldots \quad {z(0)}} \right\rbrack^{T}\quad {and}}}}{x = {\left\lbrack {{x(1)}\quad {x(2)}\quad \ldots \quad {x\left( N_{b} \right)}} \right\rbrack^{T}.}}} & (5)\end{matrix}$

[0048] N_(f) and N_(b) are the number of taps in F F F and F B Frespectively.

[0049] The combined weight vector of both the filters is given byW=[W_(f) ^(T)W_(b) ^(T)]^(T), where, W_(f)=[w_(−N) _(f) ₊₁w_(−N) _(f) ₊₂. . . w₀]^(T) and W_(b)=[1w₁ . . . w_(N) _(b) ]^(T). Thus, the estimateddata is,

{circumflex over (x)}(n)=U _(L) ^(T) W  (6)

[0050] The MSE, which is a function of filter lengths, is given by,

J _(l)(N _(f) , N _(b))=E[x(n)−{circumflex over (x)}(n)]²  (7)

[0051] The necessary and sufficient condition for the minimum of the MSEis given by $\begin{matrix}{{E\left\lbrack {U_{L}\left( {{x(n)} - {U_{L}^{T}W}} \right)} \right\rbrack} = 0} & (8)\end{matrix}$

[0052] or, equivalently, $\begin{matrix}{{{E\left\lbrack {U_{L}U_{L}^{T}} \right\rbrack}W} = {E\left\lbrack {{x(n)}U_{L}} \right\rbrack}} & (9)\end{matrix}$

[0053] If the auto-correlation matrix of the input vector U_(L) isR_(uu) and the cross correlation vector between U_(L) and the desiredresponse x(n) is p then, the optimum weights are given by,

W=R _(uu) ⁻¹ p  (10)

[0054] We evaluate the MMSE performance of the polynomial filter indetail and the final BER curves are given. The complete evaluation ofthe whole receiver including the linear filters can be found in [11].

[0055] The mean square value of the polynomial filter ε_(p)(n)=z(n)−q(n)is, $\begin{matrix}{J_{p} = {{E\left\lbrack \in_{p}^{2} \right\rbrack} = \overset{\_}{{z^{2}(n)} - {2{z(n)}{q(n)}} + {q^{2}(n)}}}} & (11)\end{matrix}$

[0056] Let us assume an independently and identically distributed (iid)data sequence x(n). Furthermore, let us assume x(n)ε{−1,1}∀n with equalprobability and no correlation between x(n) and x(m)∀m≠n. Then it can beshown that the expectation of dd powers of q(n) are zero. Expectationsof even powers depend on the channel impulse response. The results aresummarized as follows: Let us define a general symbol${{\sigma_{h}^{i}\quad {as}\quad \sigma_{h}^{i}} = {\sum\limits_{m = {- L_{b}}}^{L_{f}}{{h^{i}(m)}\quad {then}}}},$

$\begin{matrix}{{E\left\lbrack q^{i} \right\rbrack} = \left\{ \begin{matrix}{\sigma_{h}^{i}\quad {if}\quad i\quad {even}} \\{0\quad {if}\quad i\quad {odd}}\end{matrix} \right.} & (12)\end{matrix}$

[0057] where, i is a positive integer. For convenience, the timingnotation (n) is dropped in the above expression and hereafter, with anunderstanding that the manipulation is done at the discrete timeinstance n.

[0058] Now, consider the channel nonlinearity$r = {{\sum\limits_{i = 1}^{l}{A_{i}q^{i}}} + {v.}}$

[0059] We need to consider only odd order terms because the even ordernonlinearities will generate harmonics that are at least one octave awayfrom the carrier frequency. In this case r can be written as,$\begin{matrix}{r = {{\sum\limits_{i = 0}^{{({l - 1})}/2}{A_{{2i} + 1}q^{{2i} + 1}}} + v}} & (13)\end{matrix}$

[0060] Because, r has only odd power terms of q, raising r to an oddpower results in weighted sums of only odd power terms of q. Sinceexpectation of the odd power terms of q are zero,

E[r ^(i)(n)]=0 iε1,3,5,7, . . . ,∞  (14)

[0061] Here, zero mean, signal independent noise is assumed. Higher(even) order expectations of r for a given l can be computed from theexpression, $\begin{matrix}{{E\left\lbrack r^{2j} \right\rbrack} = {E\left\lbrack \left\{ {{\sum\limits_{i = 0}^{{({l - 1})}/2}{A_{{2i} + 1}q^{{2i} + 1}}} + v} \right\}^{2j} \right\rbrack}} & (15)\end{matrix}$

[0062] where, j is a positive integer.

[0063] The next task is to compute the higher order expectations of z.Using (1) and the property in (14), it can be shown that $\begin{matrix}{{E\left\lbrack z^{2} \right\rbrack} = {{\sum\limits_{i = 1}^{N}{g_{i}^{2}\sigma_{r}^{2i}}} + {\sum\limits_{\underset{{i \neq j};{{i + j} = {even}}}{i,{j = 1}}}{g_{i}g_{j}\sigma_{r}^{i + j}}}}} & (16)\end{matrix}$

[0064] Now, the task is to find${E\lbrack{zq}\rbrack} = {\sum\limits_{i = 1}^{N}{g_{i}{{E\left\lbrack {r^{i}q} \right\rbrack}.}}}$

[0065] For this, first we find the expectation of rq=r×q. Again theassumption that l is odd and noise is not correlated with the signalgives $\begin{matrix}{{E\lbrack{rq}\rbrack} = {\sum\limits_{i = 0}^{{({l - 1})}/2}{A_{{2i} + 1}\sigma_{h}^{{2i} + 2}}}} & (17)\end{matrix}$

[0066] Note that, although E[r]=0, E[rq]≠0. However, for the secondorder, E[r²q]=0, because it consists of only odd power terms of q.Similarly, all the even order expectations such as, E[r⁴q], E[r⁶q] etc.are zero. Odd order expectations can be computed from (18) where, j is apositive integer. $\begin{matrix}{{E\left\lbrack {r^{{2j} - 1}q} \right\rbrack} = {E\left\lbrack {q\left\{ {{\sum\limits_{i = 0}^{{({l - 1})}/2}{A_{{2i} + 1}q^{{2i} + 1}}} + v} \right\}^{{2j} - 1}} \right\rbrack}} & (18)\end{matrix}$

[0067] Finally, substituting all these results in (11) the mean squarederror of the polynomial filter is given by (19). Note that, in (19) theJ_(p) is a function of channel impulse response h(n), direct and inversepolynomial coefficients A_(i), g_(i) and the noise power σ_(v) ².$\begin{matrix}\begin{matrix}{J_{p} = \quad {{E\left\lbrack z^{2} \right\rbrack} - {2{E\lbrack{zq}\rbrack}} + {E\left\lbrack q^{2} \right\rbrack}}} \\{= \quad {{\sum\limits_{i = 1}^{N}{g_{i}^{2}\sigma_{r}^{2i}}} + {\underset{\underset{{i \neq j};{{i + j} = {even}}}{i,{j = 1}}}{\sum\limits^{N}}{g_{i}g_{j}\sigma_{r}^{i + j}}} -}} \\{\quad {{2{\sum\limits_{i = 1}^{N}{g_{i}{E\left\lbrack {r^{i}q} \right\rbrack}}}} + \sigma_{h}^{2}}}\end{matrix} & (19)\end{matrix}$

[0068] Let us evaluate the J_(p) under the worst and the bestconditions. Let l=1 and N=1, so that there is no nonlinearity. Then,taking g_(l)=A_(l)=1 and substituting, J_(p=σ) _(v) ². This is the bestvalue for J_(p). In all other cases J_(p)>σ_(v) ². This shows the noisepower is the lower bound of the PLF MSE, irrespective of h(n). However,as soon as l>1, J_(p) is a function of h(n) also.

[0069] In both FIG. 3 and FIG. 4 a worst case multipath dispersion isassumed. That is, all paths have equal strength. Examples are, if L=2then h=[0.5 0.5]. If L=4 then h=[0.25 0.25 0.25 0.25]. FIG. 3 shows thevariation of σ_(h) ^(k). As the order k increases the σ_(h) ^(k)exponentially decreases. The rate is high with a larger channel memorybecause of the equal gain distribution. From FIG. 4, the MSE floordecreases with the increment in the channel memory. This is because, MSEis a function of σ_(h) ^(k). As a result the HDFE compensates for thenonlinearity better when there are more paths each with small gain.Finally, FIG. 5 shows the BER performance of the HDFE and a conventionalDFE with N_(b)=L and N_(f)>>L. Even when the channel is nonlinear, theHDFE performance is very close to that of a DFE in a linear channel. Themismatch is due the PLF error.

[0070] Here we estimate both the nonlinear transfer function of the ROFlink 20 and linear impulse response h(n) of the wireless channel 22 fromthe autocovariance properties of PN sequences. Instead of using higherorder correlation functions to directly estimate higher order Volterrakernels, which lead to anomalies [9], we have used a more efficientVandermonde matrix approach to separate the Volterra kernels.Furthermore, projection of linear and nonlinear coefficients ontodifferent subspaces makes the tracking of fast changing wireless channeland the relatively static nonlinear channel virtually independent.

[0071] Simulation results show excellent estimation with just a fewtraining symbols. Since, the order of the nonlinearity is independent tothe dimension of the Vandermonde matrix, the length of the trainingsequence is independent of the order of the nonlinearity.

[0072] A model of the fiber-wireless uplink is shown in FIG. 6. Atransmit filter 38 is located in the portable unit 16. The wirelesschannel 22A is an indoor multipath channel that is modeled with a tappeddelay line filter. The nonlinear link function 24A F(.) models thecomplete optical link from the laser diode (not shown, but is at theradio access point 14) to an optical receiver (not shown) at the centralstation 10, including an RF amplifier after a photo-detector.

[0073] Assume that F(.) is continuous within the given dynamic range.Then from Weierstrass theorem, F(.) can be uniformly approximated by apolynomial of order l with an arbitrary precision ε>0. Thus,

r(n)=A ₁ q(n)+A ₂ q ²(n)+ . . . A ₁ q ¹(n)+v(n)  (20)

[0074] where, v(n) is the summation of all the optical and wirelesschannel noise. Let us define the combined linear impulse response h(n)as,

h(n)=h _(ix)(n)*c(n)  (21)

[0075] Thus, the internal signal q(n) is given as,

q(n)=x(n)*h(n)  (22)

[0076] The received signal r(n) is a nonlinear translation of q(n) plusnoise,

r(n)=F[q(n)]+v(n)  (23)

[0077] r(n) also can be written as a sum of individual higher orderterms w_(i)(n).

r(n)=w ₁+(n)+w ₂(n)+w ₃(n)+ . . . w_(l)(n)+v(n)  (24)

[0078] where, $\begin{matrix}{{w_{1}(n)} = {A_{1}{\sum\limits_{m = {- \infty}}^{\infty}{{h(m)}{x\left( {n - m} \right)}}}}} & (25)\end{matrix}$

$\begin{matrix}{{w_{2}(n)} = {A_{2}{\sum\limits_{m_{1} = {- \infty}}^{\infty}{{h\left( m_{1} \right)}{x\left( {n - m_{1}} \right)}{\sum\limits_{m_{2} = {- \infty}}^{\infty}{{h\left( m_{2} \right)}{x\left( {n - m_{2}} \right)}}}}}}} & (26)\end{matrix}$

[0079] Finally, $\begin{matrix}{{w_{l}(n)} = {A_{l}{\sum\limits_{m_{1} = {- \infty}}^{\infty}\quad {\ldots \quad {\sum\limits_{m_{1} = {- \infty}}^{\infty}{\prod\limits_{i = 1}^{l}\quad {{h\left( m_{i} \right)}{x\left( {n - m_{i}} \right)}}}}}}}} & (27)\end{matrix}$

[0080] Expressing in this form facilitates separation of correspondingVolterra kernels [9]. Let us define the i^(th) order separable Volterrakernel g_(i) as, $\begin{matrix}{{g_{i}\left( {m_{1},m_{2},\ldots \quad,m_{i}} \right)} = {A_{i}{\sum\limits_{\tau = {- \infty}}^{\infty}{\prod\limits_{i = 1}^{i}\quad {h\left( {m_{j} - \tau} \right)}}}}} & (28)\end{matrix}$

[0081] This general definition includes the memory term τ. However inour model we assume no memory for F(.). Using this kernel description,the general term w_(i)(1≦i≦1) can be written as, $\begin{matrix}{{{w_{i}(n)} = {\sum\limits_{m_{i} = {- \infty}}^{\infty}\quad {\ldots \quad {\sum\limits_{m_{i} = {- \infty}}^{\infty}{g_{i}\left( {m_{1},m_{2},\ldots \quad,m_{i}} \right)}}}}}\quad {\prod\limits_{j = 1}^{i}\quad {{x\left( {n - m_{j}} \right)}.}}} & (29)\end{matrix}$

[0082] Therefore, for a given transmitted sequence x(n), the receivedsignal r(n) consists of higher order terms of the present and previoussamples that are multiplied with each other.

[0083] From equations (29) and (24), the received signal r(n) can bewritten as, $\begin{matrix}{{{r(n)} = {\sum\limits_{i = 1}^{l}{\sum\limits_{m_{i} = {- \infty}}^{\infty}\quad {\ldots \quad {\sum\limits_{m_{i} = {- \infty}}^{\infty}{g_{i}\left( {m_{1},m_{2},{\ldots \quad m_{i}}} \right)}}}}}}\quad \quad {{\prod\limits_{j = 1}^{i}\quad {x\left( {n - m_{j}} \right)}} + {v(n)}}} & (30)\end{matrix}$

[0084] where, x(n) is a PN sequence of length N_(c). As shown previouslyr(n) also can be written simply as, $\begin{matrix}{{r(n)} = {{\sum\limits_{i = 1}^{l}{w_{i}(n)}} + {v(n)}}} & (31)\end{matrix}$

[0085] Let rr(n)=r(n)−{overscore (r(n))} and xx(n)=x(n)−{overscore(x(n))} that are zero mean processes. The cross covariance of r and x,

_(rx) is computed as,

_(rx)(σ)={overscore (rr(n)xx(n−σ))}  (32)

[0086] The zero mean process rr(n) can now be written as,${{{rr}(n)} = {\sum\limits_{i = 1}^{l}{\sum\limits_{m_{1} = {- \infty}}^{\infty}\quad {\ldots \quad {\sum\limits_{m_{i} = {- \infty}}^{\infty}{g_{i}\left( {m_{1},m_{2},{\ldots \quad m_{i}}} \right)}}}}}}\quad$

$\begin{matrix}{\left\lbrack {{\prod\limits_{i = 1}^{i}\quad {x\left( {n - m_{j}} \right)}} - {\prod\limits_{i = 1}^{i}\quad \overset{\_}{x\left( {n - m_{j}} \right)}}} \right\rbrack + {v(n)}} & (33)\end{matrix}$

[0087] (If we assume no correlation between the additive noise v(n) andthe input sequence x(n), then the covariance function

_(rx)(σ) can be written as, $\begin{matrix}{{\Re_{rx}(\sigma)} = \frac{\sum\limits_{i = 1}^{l}{\sum\limits_{m_{1} = {- \infty}}^{\infty}\quad {\ldots \quad {\sum\limits_{m_{i} = {- \infty}}^{\infty}{{g_{i}\left( {m_{1},m_{2},{\ldots \quad m_{i}}} \right)}\quad.}}}}}{\left\lbrack {{\prod\limits_{j = 1}^{i}\quad {x\left( {n - m_{j}} \right)}} - {\prod\limits_{j = 1}^{i}\quad \overset{\_}{x\left( {n - m_{j}} \right)}}} \right\rbrack\left\lbrack {{x\left( {n - \sigma} \right)} - \overset{\_}{\left. {x\left( {n - \sigma} \right)} \right\rbrack}} \right.}} & (34)\end{matrix}$

[0088] Equation (34) above, is actually a summation of the crosscorrelations of xx(n) with each w_(i)(n) of rr(n). This can be simplywritten as, $\begin{matrix}{{\Re_{rx}(\sigma)} = {\sum\limits_{i = 1}^{l}{\Re_{w_{i}x}(\sigma)}}} & (35)\end{matrix}$

[0089] However, if

_(rx)(σ) is evaluated directly as defined above, it leads to anomalies([9], [13]). This is because,

_(rx)(σ) involves higher order correlation functions of pseudorandomsequences, which is not always computable [8]. This problem can beavoided by isolating the first order covariance function

_(wr) ₁ _(x)(σ) from the rest by some means.

[0090] For multilevel transmission, the easiest approach is to repeatthe training a few (say N_(t)) times with different amplitudes. Thisapproach was originally proposed by Billings [9]. This procedureeffectively generates a system of N_(t) simultaneous equations insteadof a single equation which, enables solving for the

_(w) ₁ _(x)(σ) without computing higher order correlation functions. Letus consider multilevel input signals α_(i)x(n) where, α_(i)≠α_(j)∀i≠jThen, $\begin{matrix}{{{\Re_{r_{a_{i}}x}(\sigma)} = {{\sum\limits_{j = 1}^{N_{i}}{\alpha_{i}^{j}{\Re_{w_{j}x}(\sigma)}\quad i}} = 1}},2,\ldots \quad,N_{t}} & (36)\end{matrix}$

[0091] Note that N_(t) does not have to be the same as the order of thenonlinearity, since N_(t) is not directly used to compute the polynomialcoefficients. The value of N_(t) can be as low as 2 or 3. The simulationshows excellent results with N_(t)=3. Expanding equation (36) in matrixform gives, $\begin{matrix}{\begin{bmatrix}{\Re_{r\quad \alpha_{\quad_{1}}x}(\sigma)} \\{\Re_{r\quad \alpha_{\quad_{2}}x}(\sigma)} \\\vdots \\{\Re_{r\quad \alpha_{\quad_{Nt}}x}(\sigma)}\end{bmatrix} = {\begin{bmatrix}\alpha_{1} & \alpha_{1}^{2} & \ldots & \alpha_{1}^{N_{t}} \\\alpha_{2} & \alpha_{2}^{2} & \ldots & \alpha_{2}^{N_{t}} \\\vdots & \vdots & \vdots & \vdots \\{\alpha_{N}}_{t} & \alpha_{N_{t}}^{2} & \ldots & \alpha_{N_{t}}^{N_{t}}\end{bmatrix}\begin{bmatrix}{\Re_{w_{\quad_{1}}x}(\sigma)} \\{\Re_{w_{\quad_{2}}x}(\sigma)} \\\vdots \\{\Re_{w_{\quad_{Nt}}x}(\sigma)}\end{bmatrix}}} & (37)\end{matrix}$

[0092] The coefficient matrix α can be re-written as, $\begin{bmatrix}\alpha_{1} & 0 & \ldots & 0 \\0 & \alpha_{2} & \ldots & 0 \\0 & 0 & \ldots & \ldots \\\ldots & \ldots & \cdot & \ldots \\0 & \ldots & \ldots & \alpha_{N_{t}}\end{bmatrix}\begin{bmatrix}1 & \alpha_{1} & \alpha_{1}^{2} & \ldots & \alpha_{1}^{N_{t} - 1} \\1 & \alpha_{2} & \alpha_{2}^{2} & \ldots & \alpha_{2}^{N_{t} - 1} \\\vdots & \vdots & \vdots & \vdots & \vdots \\1 & {\alpha_{N}}_{t} & \alpha_{N_{t}}^{2} & \ldots & \alpha_{N_{t}}^{N_{t} - 1}\end{bmatrix}$

[0093] The first diagonal matrix above, is non-singular for all α_(i)≠0.The second matrix is the well known Vandermonde matrix. The determinantof the Vandermonde matrix is given below, which is non zero forα_(i)≠α_(j)$\prod\limits_{1 \leq i \leq j \leq N_{t}}\left( {\alpha_{i} - \alpha_{j}} \right)$

[0094] Thus for every value of α, equation (36) has a unique solutionfor

_(w) _(i) _(x)(σ); i=1,2, . . . , N_(t)

[0095] Now from equations (6),

_(w) ₁ x(σ) can be written as $\begin{matrix}{{\Re_{w_{i}x}(\sigma)} = {A_{1}{\sum\limits_{m = {- \infty}}^{\infty}{{h(m)}{\Re_{xx}\left( {\sigma - m} \right)}}}}} & (38)\end{matrix}$

[0096] where,

_(xx)(σ) is the auto-covariance of the PN sequence x(n) that is definedas [8], $\begin{matrix}{{\Re_{xx}(\sigma)} = {\sum\limits_{m_{1} = 0}^{N_{c} - 1}{{x(n)}{x\left( {n + \sigma} \right)}}}} & (39)\end{matrix}$

[0097] The function

_(xx)(σ) is periodic and can be easily determined if it is a maximallength sequence. In this case, since x(n) has unit amplitude,$\begin{matrix}{{\Re_{xx}(\sigma)} = \left\{ \begin{matrix}{\quad {{1\quad {if}\quad \sigma} = {0\quad {mod}\quad N_{c}}}} \\{{{{- 1}/N_{c}}\quad {if}\quad \sigma} \neq {0\quad {mod}\quad N_{c}}}\end{matrix} \right.} & (40)\end{matrix}$

[0098] Assume chip level synchronization is achieved and N_(c)>>1. Ifthe correlation is computed within the time period 0≦n≦N_(c)−1, then

_(xx)(σ) can be written as

_(xx)(σ)=g(σ),

[0099] Therefore, the equation (38) simplifies to, $\begin{matrix}{{\Re_{w_{1}x}(\sigma)} = {A_{1}{\sum\limits_{m = 0}^{N_{c} - 1}{{h(m)}{\delta \left( {\sigma - m} \right)}}}}} & (41)\end{matrix}$

[0100] Using the convolution properties of the impulse function,

_(w) ₁ _(x)(σ)=A ₁ h(m)  (42)

[0101] Using equation (42), the impulse response of the unknown linearchannel h(n) multiplied with the linear gain A₁ of the nonlinear channelcan be computed. Note that the length of the PN sequence, must be largerthan the channel memory for a complete identification of h(n).

[0102] Having identified the linear part of the Wiener system, the taskis to identify the nonlinear part. Referring to FIG. 2, the transmittedtraining sequence x(n) is known; the impulse response of the linear parth(n) has been estimated as shown in the last section. Thus, the unknowninternal signal q(n)=x(n)*h(n), can be estimated. The output of thenonlinear part r(n) is known to the receiver. Now, the task is to use anappropriate curve fitting algorithm such as a least squares polynomialfit to estimate the polynomial coefficients A_(i)(1≦i≦1) of thenonlinear part F(.).

[0103] In the least squares curve fitting method, the goal is to fit aset of sample points to a polynomial, so that the squared error betweenthe actual sample points and the polynomial estimate is minimized. Thismethod requires the order of the polynomial l to be smaller than thenumber of sample points N_(L), which is a very loose condition (usuallyl<<N_(L)). The problem is equivalent to solving an over determinedsystem with N_(L) equations and l unknowns in the least squares sense.Here, N_(L)=N_(c)*N_(t).

[0104] Let the estimated signal {circumflex over (r)}(n) be given as afunction of the estimated coefficients Â_(i),

{circumflex over (r)}(n)=Â ₁ q(n)+Â ₂ q ²(n)+ . . . Â ₁ q¹(n)+v(n)  (43)

[0105] Now, the target is to find a polynomial coefficient vector Â oflength (l+1) in a least squares sense such that, $\begin{matrix}{\hat{A} = {\arg\left\lbrack {\min\limits_{\hat{A}}{\sum\limits_{i = 1}^{N_{L}}\left\lbrack {r_{i} - {{\hat{r}}_{i}\left( {q_{i};{{\hat{A}}_{0}\quad \ldots \quad {\hat{A}}_{l}}} \right\rbrack}^{2}} \right\rbrack}} \right.}} & (44)\end{matrix}$

[0106] Let us define vectors q and r of length N_(L), that are made upof the signals q(n) and r(n) respectively. Also, let us define aVandermonde matrix V, such that each row of V is a polynomial of thecorresponding data point in q. V has N_(L) rows and (l+1) columns and isgiven by, $\begin{matrix}{V = \begin{bmatrix}{q^{l}(1)} & {q^{l - 1}(1)} & \ldots & {q(1)} & 1 \\{q^{l}(2)} & {q^{l - 1}(2)} & \ldots & {q(2)} & 1 \\\ldots & \ldots & \ldots & \ldots & 1 \\\ldots & \ldots & \ldots & \ldots & \ldots \\{q^{l}\left( N_{L} \right)} & {q^{l - 1}\left( N_{L} \right)} & \ldots & {q\left( N_{L} \right)} & 1\end{bmatrix}} & (45)\end{matrix}$

[0107] Minimization of the sum of squared errors in equation (44)yields,

VÂ={circumflex over (r)}  (46)

[0108] An efficient way of computing Â is performing an orthogonaltriangular decomposition of the Vandermonde matrix V so that it can bewritten as,

V=QR  (47)

[0109] where, R is an upper triangular matrix and Q is an orthonormalmatrix that has the same dimension as V. The above decomposition yields,

QPÂ={circumflex over (r)}  (48)

[0110] Since the unitary matrix Q has the property that Q^(T)=Q⁻¹, thissimplifies to,

RÂ=Q ^(T) {circumflex over (r)}  (49)

[0111] Note that Q^(T)r is a vector of length (l+1). Thus, since R is atriangular matrix, this equation can be easily solved for Â by backsubstitution. The mean error between the actual data r(n) and thecomputed data is given by:

ε=r−VÂ  (50)

[0112] Finally, the order of the polynomial l and number anddistribution of the data ts has to be selected to minimize the meansquared error.

[0113] To evaluate the algorithm, a simulation was run by assumingdifferent channel lse responses and different nonlinear characteristics.The Simulink™ in Matlab™ environment is used for simulation.

[0114] The block diagram used for simulation is shown in FIG. 7. A PNsequence x(n) is rated from the generator block. The gain blocksimulates the input level coefficient The linear system is a discretetap-delay line filter. The nonlinear system is modeled polynomialfunction. An infinite signal to noise ratio is used. The received signalr(n) is stored at a buffer of memory N_(c). N_(c) is taken as 31. Thecross correlation is performed after removing the mean of the bufferedr(n) and x(n). The output is

h(n)=δ(n)−0.8δ(n−7)+0.6δ(n−13)−0.4δ(n−17)   (51)

h(n)=δ(n)−0.8δ(n−5)+0.6δ(n−9)−0.4δ(n−11)+0.3δ(n−14)−0.5δ(n−17)+0.4δ(n−21)+0.1δ(n−25)  (52)

[0115] This is stored and the simulation is repeated with a different avalue. The values used were α=1,1.2 and 1.4. Then

_(w) _(j) _(x)(σ) is calculated using the previously derivedrelationship.

[0116] Two different impulse responses h(n) are used. One with fourpaths and the other with eight paths. These are given below,

h(n)=δ(n)−0.8δ(n−7)+0.6δ(n−13)−0.4δ(n−17)  (51)

[0117]h(n)=δ(n)−0.8δ(n−5)+0.6δ(n−9)−0.4δ(n−11)+0.3δ(n−14)−0.5δ(n−17)+0.4δ(n−21)+0.1δ(n−25)  (52)

[0118] The original impulse response and the estimated impulse responsesare shown in FIG. 8 and FIG. 9 respectively.

[0119] Comparing the theoretical and simulation results, it can be seenthat the proposed algorithm is very efficient in identifying Wiener typenonlinear systems. The number of levels a is independent of the order ofnonlinearity. We used 20 symbols for training which, gives pretty goodresults. Even fewer symbols are enough depending on the accuracyrequired. Ideally (when there is no noise and with perfect numericalprecision), one symbol is enough for identification. Here, the samplingrate is the same as the chip rate.

[0120] Thus, the resolution of the impulse response depends on the chiptime. However, the memory of the channel can be as long as the length ofthe PN sequence.

[0121] Identification of the linear and nonlinear systems are found tobe quite independent. Two different nonlinear systems are used for thesimulation. A fourth order (even) system,

r(n)=−0.6q ⁴(n)+1.2q ²(n)  (53)

[0122] and a third order (odd) system,

r(n)−0.35q ³(n)+q(n)  (54)

[0123] The received symbols r(n) and the estimated curves, using theorthogonal triangular decomposition algorithm, are shown in FIG. 10. Theestimated polynomials from the received data using the decompositionalgorithm are,

r(n)=−0.61q ⁴(n)+1.19q ²(n)  (55)

r(n)=−0.35q ³(n)+0.99q(n)  (56)

[0124] These are very close to the original polynomials.

[0125] It is important to mention that the shape of the nonlinearity isimmaterial for the algorithm to work. It always identifies the exactcurve. However, we need sample points that span the whole dynamic rangeof interest. This is better achieved when the linear channel has severemultipath conditions because, the received symbols will then exhibitlarge amplitude fluctuations. If this is not the case then we may haveto increase the number of levels α, so that the entire dynamic range iscovered. The computational complexity of the nonlinear identificationdepends on the number of received sample points N_(t) and the order ofthe polynomial l.

[0126] As seen from the foregoing, we have described here a uniqueHammerstein type decision-feedback-equalizer (HDFE) to compensate fordistortions of the combined fiber-wireless uplink (FIG. 2.). This is thefirst time any such attempt is made. The receiver has two parts. First,it estimates both the fiber and the wireless channels individually usingauto-covariance properties of PN sequences. The complete algorithm isdescribed above. Then, it performs compensation for this combinedfiber-wireless channel. This is also described above.

[0127] The HDFE has the following advantages: It compensates for thenonlinearity of wS the whole ROF link, not just the laser diode(contrast to some previous work). The compensation is done at baseband,so that the clock speed of the DSP chip can be slower. Being a basebandscheme, the receiver architecture is independent of the radio frequency.The receiver architecture is independent of ROF link parameters likewavelength and type of optical modulation used, as long as thememoryless assumption holds. This is an adaptive scheme. Therefore, thereceiver adapts itself to compensate for different ROF links. Thereceiver tracks any modification or drift in the ROF link parameters.The receiver architecture separates the compensation of the fastchanging wireless channel from the relatively static fiber channel. Thishas the advantage that, the polynomial filter coefficients have to beupdated only occasionally compared to linear filter coefficients. In amultiuser environment, everybody shares the same fiber channel but eachuser has different wireless channel. In this case, a single polynomialfilter is sufficient for all users, only the linear filters have to bedifferent for different users. The BER performance of the HDFE in thefiber-wireless channel is close to that of a standard DFE in a linearchannel.

[0128] Finally, the HDFE equalizer is applicable to any Wiener typenonlinear channel. A Wiener system has a time dispersive system such asa wireless channel or a coaxial cable followed by a static memorylessnonlinear system. These types of channels are common in digitalcommunications.

REFERENCES

[0129] [1] Bob Davies, Optical Single Sideband for Broadband andSubcarrier Systems, Ph.D. thesis, University of Alberta, 2000

[0130] [2] Raziq Pervez and Masao Nakagawa. “Semiconductor laser'snonlinearity compensation for DS=CDMA optical transmission system bypost nonlinearity recovery. Block,” IECIE Transactions onCommunications, vol. E-79 B, no. 3, March 1996.

[0131] [3] Xavier Fernando and Abu Sesay, “Nonlinear distortioncompensation of microwave fiber optic links with asymmetric adaptivefilters,” in Proceedings of the IEEE International Microwave Symposium,June 2000.

[0132] [4] Xavier Fernando and Abu Sesay, “Nonlinear channel estimationusing correlation properties of PN sequences,” in Proceedings of theCanadian Conference on Electrical and Computer Engineering), Toronto,ON, May 2001.

[0133] [5] Xavier Fernando and Abu Sesay, “Higher order adaptive filtercharacterization of microwave fiber optic link nonlinearity,” inProceedings of the SPIE, The International Society of for OpticalEngineering, January 2000, vol. 3927, pp. 39-49.

[0134] [6] Xavier Fernando and Abu Sesay, “Higher order adaptive filterbased predistortion for nonlinear distortion compensation of radio overfiber links,” in Proceedings of the IEEE International Conference onCommunications, June 2000.

[0135] [7] Norbert wiener, Nonlinear Problems in Random Theory,Technology press of MIT and John Wiley and Sons Inc. New York, 1958.

[0136] [8] D. V. Sarwate and M. B. Pursley, ‘Crosscorrelation propertiesof pseudorandom and related sequences,” Proceedings of the IEEE, 1980,

[0137] [9] S. A. Billings and S. Y. Fakhouri, “Identification ofnonlinear systems using correlation analysis of pseudorandom inputs,”Int. J. Systems Science, 1980

[0138] [10] C. A. Belfiore and J. H. Park Jr., “Decision feedbackequalization,” Proceedings of the IEEE, vol. 67, no. 8, pp. 1143-1156,1979.

[0139] [11] Xavier Fernando, An Optical Fiber Based Wireless AccessScheme with Asymmetry, Phd. Thesis, University of Calgary, 2001

[0140] [12] S. A. Billings and S. Y. Fakhouri, “Identification ofnonlinear systems using the wiener model,” Electronic letters, 1977

[0141] [13] S. A. Billings and S. Y. Fakhouri, “Identification ofsystems containing linear dynamic and static nonlinear elements,”Automatica, 1982.

[0142] [14] Simon Haykin, Adaptive Filter Theory, Prentice-Hall, NewJersey, 2002 (pages 513-520)

[0143] All references cited herein are incorporated by reference.

[0144] Immaterial modifications may be made to the invention describedhere without departing from the essence of the invention.

We claim:
 1. In a central base station of a communications network, wherein the central base station communicates over a channel with multiple portable units, the channel having a channel impulse response and the channel contributing linear and nonlinear distortion to a signal x(n) transmitted over the channel, an equalizer comprising: a polynomial filter for receiving signals from the portable units and for producing an output signal z(n) that is compensated for nonlinear channel distortion; and a filter section following the polynomial filter for compensating for linear channel distortion.
 2. In the central base station of claim 1, the polynomial filter being configured to apply a polynomial having polynomial coefficients gi calculated from the signals received from the portable units.
 3. In the central base station of claim 2, the polynomial coefficients being calculated without explicitly estimating a polynomial that characterizes the channel.
 4. In the central base station of claim 3, the equalizer being configured to calculate the polynomial coefficients gi by: estimating the channel impulse response h(n); correlating the channel impulse response h(n) with the signal x(n) transmitted over the channel to produce a signal q(n); and selecting the coefficients gi so that the output z(n) of the polynomial filter is proportional to q(n).
 5. In the central base station of claim 4, the coefficients of the polynomial being selected by a method from the group consisting of QR decomposition and an adaptive technique.
 6. In the central base station of claim 5, the coefficients of the polynomial being selected by QR decomposition.
 7. In the central base station of claim 1, the polynomial filter being memoryless.
 8. In the central base station of claim 1, the filter section being a decision feedback equalizer having a feedforward filter and a feedback filter.
 9. In the central base station of claim 1, the equalizer being configured to estimate the channel impulse response h(n) by using correlation properties of PN sequences.
 10. In the central base station of claim 9, the correlation properties being determined using a Vandermonde matrix approach with projection of linear and nonlinear coefficients of the feedforward filter and the feedback filter onto different subspaces.
 11. In the central base station of claim 1, the equalizer being configured to adapt to a channel that incorporates a radio link that contributes a linear distortion to the channel.
 12. In the central base station of claim 11, the equalizer being configured to adapt to a channel that incorporates an optical fiber link that contributes a nonlinear distortion to the channel.
 13. In the central base station of claim 4, the equalizer being a baseband equalizer. 